A method for static aeroelastic trim analysis and flight loads computation of a flexible aircraft with large deformations has been presented in this paper,which considers the geometric nonlinearity of the structure an...A method for static aeroelastic trim analysis and flight loads computation of a flexible aircraft with large deformations has been presented in this paper,which considers the geometric nonlinearity of the structure and the nonplanar effects of aerodynamics.A nonplanar vortex lattice method is used to compute the nonplanar aerodynamics.The nonlinear finite element method is introduced to consider the structural geometric nonlinearity.Moreover,the surface spline method is used for structure/aerodynamics coupling.Finally,by combining the equilibrium equations of rigid motions of the deformed aircraft,the nonlinear trim problem of the flexible aircraft is solved by iterative method.For instance,the longitudinal trim analysis of a flexible aircraft with large-aspect-ratio wings is carried out by both the nonlinear method presented and the linear method of MSC Flightloads.Results obtained by these two methods are compared,and it is indicated that the results agree with each other when the deformation is small.However,because the linear method of static aeroelastic analysis does not consider the nonplanar aerodynamic effects or structural geometric nonlinearity,it is not applicable as the deformations increase.Whereas the nonlinear method presented could solve the trim problem accurately,even the deformations are large,which makes the nonlinear method suitable for rapid and efficient analysis in engineering practice.It could be used not only in the preliminary stage but also in the detail stage of aircraft design.展开更多
Vortices and bound states offer an effective means of comprehending the electronic properties of superconductors.Recently,surface-dependent vortex core states have been observed in the newly discovered kagome supercon...Vortices and bound states offer an effective means of comprehending the electronic properties of superconductors.Recently,surface-dependent vortex core states have been observed in the newly discovered kagome superconductors CsV_(3)Sb_(5).Although the spatial distribution of the sharp zero energy conductance peak appears similar to Majorana bound states arising from the superconducting Dirac surface states,its origin remains elusive.In this study,we present observations of tunable vortex bound states(VBSs)in two chemically-doped kagome superconductors Cs(V_(1-x)Tr_(x))_(3)Sb_(5)(Tr=Ta or Ti),using low-temperature scanning tunneling microscopy/spectroscopy.The CsV_(3)Sb_(5)-derived kagome superconductors exhibit full-gap-pairing superconductivity accompanied by the absence of long-range charge orders,in contrast to pristine CsV_(3)Sb_(5).Zero-energy conductance maps demonstrate a field-driven continuous reorientation transition of the vortex lattice,suggesting multiband superconductivity.The Ta-doped CsV_(3)Sb_(5)displays the conventional cross-shaped spatial evolution of Caroli-de Gennes-Matricon bound states,while the Tidoped CsV_(3)Sb_(5)exhibits a sharp,non-split zero-bias conductance peak(ZBCP)that persists over a long distance across the vortex.The spatial evolution of the non-split ZBCP is robust against surface effects and external magnetic field but is related to the doping concentrations.Our study reveals the tunable VBSs in multiband chemically-doped CsV_(3)Sb_(5)system and offers fresh insights into previously reported Y-shaped ZBCP in a non-quantum-limit condition at the surface of kagome superconductor.展开更多
Experimental folding fin models with an adjustable free-play are tested in a wind tunnel.The fin structure is modeled using the free-interface component mode synthesis method,and its free-play is modeled as four indep...Experimental folding fin models with an adjustable free-play are tested in a wind tunnel.The fin structure is modeled using the free-interface component mode synthesis method,and its free-play is modeled as four independent nonlinear springs with asymmetric stiffness.A nonplanar unsteady vortex-lattice method considering compressibility is employed to address nonlinear deformation and high subsonic flow.Surface spline interpolation is improved through projection and partition.The aeroelastic characteristics of folding fins with different free-play magnitudes,initial conditions and elastic-axis positions are analyzed using an established time-marching method because of its relatively small computation scale and high precision.The results show good consistency among the presented method,the wind tunnel test and the harmonic balance method.There is a negative correlation between the critical speed of divergent motion and the ratio of the initial condition to the free-play magnitude.If either the free-play magnitude or the initial condition is extreme(tiny or vast),the system nonlinearity degenerates to linearity.Generally,the flutter prevention design of a linear model can be applied to a nonlinear model,such as moving the elastic-axis position aftward.The presented fin configuration exhibits an unstable limit cycle oscillation because the orders of coupled flutter modes do not change with variations in equivalent linear stiffness.展开更多
In this work, we study approximations of supercritical or suction vortices in tornadic flows and their contribution to tornadogenesis and tornado maintenance using self-avoiding walks on a cubic lattice. We extend the...In this work, we study approximations of supercritical or suction vortices in tornadic flows and their contribution to tornadogenesis and tornado maintenance using self-avoiding walks on a cubic lattice. We extend the previous work on turbulence by A. Chorin and collaborators to approximate the statistical equilibrium quantities of vortex filaments on a cubic lattice when both an energy and a statistical temperature are involved. Our results confirm that supercritical (smooth, “straight”) vortices have the highest average energy and correspond to negative temperatures in this model. The lowest-energy configurations are folded up and “balled up” to a great extent. The results support A. Chorin’s findings that, in the context of supercritical vortices in a tornadic flow, when such high-energy vortices stretch, they need to fold and transfer energy to the surrounding flow, contributing to tornado maintenance or leading to its genesis. The computations are performed using a Markov Chain Monte Carlo approach with a simple sampling algorithm using local transformations that allow the results to be reliable over a wide range of statistical temperatures, unlike the originally used pivot algorithm that only performs well near infinite temperatures. Efficient ways to compute entropy are discussed and show that a system with supercritical vortices will increase entropy by having these vortices fold and transfer their energy to the surrounding flow.展开更多
We present an overview of our recent theoretical studies on the quantum phenomena of the spin-1 Bose Einstein condensates, including the phase diagram, soliton solutions and the formation of the topological spin textu...We present an overview of our recent theoretical studies on the quantum phenomena of the spin-1 Bose Einstein condensates, including the phase diagram, soliton solutions and the formation of the topological spin textures. A brief exploration of the effects of spin-orbit coupling on the ground-state properties is given. We put forward proposals by using the transmission spectra of an optical cavity to probe the quantum ground states： the ferromagnetic and polar phases. Quasi-one-dimension solitons and ring dark solitons are studied. It is predicted that characteristics of the magnetic solitons in optical lattice can be tuned by controlling the long-range light-induced and static magnetic dipole- dipole interactions; solutions of single-component magnetic and single-, two-, three-components polar solitons are found; ring dark solitons in spin-1 condensates are predicted to live longer lifetimes than that in their scalar counterparts. In the formation of spin textures, we have considered the theoretical model of a rapidly quenched and fast rotating trapped spin-1 Bose Einstein condensate, whose dynamics can be studied by solving the stochastic projected Gross-Pitaevskii equations. Spontaneous generation of nontrivial topological defects, such as the hexagonal lattice skyrmions and square lattice of half-quantized vortices was predicted. In particular, crystallization of merons （half skyrmions） can be generated in the presence of spin-orbit coupling.展开更多
One of the main issues concerning the standard Vortex Lattice Method is its application to partially or fully detached flow conditions,where non-linear aerodynamic characteristics appear as the angle of attack increas...One of the main issues concerning the standard Vortex Lattice Method is its application to partially or fully detached flow conditions,where non-linear aerodynamic characteristics appear as the angle of attack increases and/or the aspect ratio decreases.In order to solve such limitations,a pure numerical approach based entirely on the Vortex Lattice Method concepts has been developed.The so-called steady“Full Multi-wake Vortex Lattice Method”comes from the main hypothesis that each discretized element on the body’s surface detaches their own wakes downstream.The obtained results match for lift,drag and moment coefficients for the entire aspect ratio range configurations(under straight wakes and inviscid assumptions).Future unsteady versions of such a multi-wake approach could improve the current results obtained through Vortex Element Methods(as vortons or isolated vortex filaments).展开更多
Engineering of the orbital angular momentum(OAM)of light due to interaction with photonic lattices reveals rich physics and motivates potential applications.We report the experimental creation of regularly distributed...Engineering of the orbital angular momentum(OAM)of light due to interaction with photonic lattices reveals rich physics and motivates potential applications.We report the experimental creation of regularly distributed quantized vortex arrays in momentum space by probing the honeycomb and hexagonal photonic lattices with a single focused Gaussian beam.For the honeycomb lattice,the vortices are associated with Dirac points.However,we show that the resulting spatial patterns of vortices are strongly defined by the symmetry of the wave packet evolving in the photonic lattices and not by their topological properties.Our findings reveal the underlying physics by connecting the symmetry and OAM conversion and provide a simple and efficient method to create regularly distributed multiple vortices from unstructured light.展开更多
Electric vertical take-off and landing(eVTOL)aircraft with multiple lifting rotors or prop-rotors have received significant attention in recent years due to their great potential for next-generation urban air mobility...Electric vertical take-off and landing(eVTOL)aircraft with multiple lifting rotors or prop-rotors have received significant attention in recent years due to their great potential for next-generation urban air mobility(UAM).Numerical models have been developed and validated as predictive tools to analyze rotor aerodynamics and wake dynamics.Among various numerical approaches,the vortex method is one of the most suitable because it can provide accurate solutions with an affordable computational cost and can represent vorticity fields downstream without numerical dissipation error.This paper presents a brief review of the progress of vortex methods,along with their principles,advantages,and shortcomings.Applications of the vortex methods for modeling the rotor aerodynamics and wake dynamics are also described.However,the vortex methods suffer from the problem that it cannot deal with the nonlinear aerodynamic characteristics associated with the viscous effects and the flow behaviors in the post-stall regime.To overcome the intrinsic drawbacks of the vortex methods,recent progress in a numerical method proposed by the authors is introduced,and model validation against experimental data is discussed in detail.The validation works show that nonlinear vortex lattice method(NVLM)coupled with vortex particle method(VPM)can predict the unsteady aerodynamic forces and complex evolution of the rotor wake.展开更多
This paper presents the application of an artificial neural network to develop an approach to determine and study the energy-optimal wing kinematics of a hovering bionic hawkmoth model.A three-layered artificial neura...This paper presents the application of an artificial neural network to develop an approach to determine and study the energy-optimal wing kinematics of a hovering bionic hawkmoth model.A three-layered artificial neural network is used for the rapid prediction of the unsteady aerodynamic force acting on the wings and the required power.When this artificial network is integrated into genetic and simplex algorithms,the running time of the optimization process is reduced considerably.The validity of this new approach is confirmed in a comparison with a conventional method using an aerodynamic model based on an extended unsteady vortex-lattice method for a sinu soidal wing kinematics problem.When studying the obtained results,it is found that actual hawkmoths do not hover under an energy-optimal condition.Instead,by tilting the stroke plane and lowering the wing positions,they can compromise and expend some energy to enhance their maneuverability and the stability of their flight.展开更多
Here we review recent small-angle scattering studies of the vortex lattice in a range of type-II superconductors carried out by our group. Emphasis is placed on providing examples of the kind of information which can ...Here we review recent small-angle scattering studies of the vortex lattice in a range of type-II superconductors carried out by our group. Emphasis is placed on providing examples of the kind of information which can be obtained by such measurements, focusing in particular on studies of the vortex lattice structure and form factor in LuNi2B2C, TmNi2B2C, CeCoIn5 and Ba（Fe0.93Co0.07）2As2.展开更多
The standard Ginzburg-Landau (GL) equations are only valid in the vicinity of the critical temperature. Based on the Eilenberger equations for a single band and s-wave superconductor, we derive a modified version of t...The standard Ginzburg-Landau (GL) equations are only valid in the vicinity of the critical temperature. Based on the Eilenberger equations for a single band and s-wave superconductor, we derive a modified version of the standard GL equations to improve the applicability of the standard formalism at temperature away from the critical temperature. It is shown that in comparison with previous studies, our method is more convenient to calculate and our modified equations are also compatible with a dirty superconductor. To illustrate the usefulness of our formalism, we solve the modified equations numerically and give the magnetic field distribution in the mixed state at any temperature. The results show that the vortex lattice could be still observed even away from the critical temperature (e.g., T/Tc = 0.3).展开更多
基金supported by the National Natural Science Foundation of China (Grant Nos. 11172025,91116005)the Research Fund for the Doctoral Program of Higher Education of China (Grant No.20091102110015)
文摘A method for static aeroelastic trim analysis and flight loads computation of a flexible aircraft with large deformations has been presented in this paper,which considers the geometric nonlinearity of the structure and the nonplanar effects of aerodynamics.A nonplanar vortex lattice method is used to compute the nonplanar aerodynamics.The nonlinear finite element method is introduced to consider the structural geometric nonlinearity.Moreover,the surface spline method is used for structure/aerodynamics coupling.Finally,by combining the equilibrium equations of rigid motions of the deformed aircraft,the nonlinear trim problem of the flexible aircraft is solved by iterative method.For instance,the longitudinal trim analysis of a flexible aircraft with large-aspect-ratio wings is carried out by both the nonlinear method presented and the linear method of MSC Flightloads.Results obtained by these two methods are compared,and it is indicated that the results agree with each other when the deformation is small.However,because the linear method of static aeroelastic analysis does not consider the nonplanar aerodynamic effects or structural geometric nonlinearity,it is not applicable as the deformations increase.Whereas the nonlinear method presented could solve the trim problem accurately,even the deformations are large,which makes the nonlinear method suitable for rapid and efficient analysis in engineering practice.It could be used not only in the preliminary stage but also in the detail stage of aircraft design.
基金supported by the National Natural Science Foundation of China(61888102,52022105,92065109,and 12174428)the National Key Research and Development Projects of China(2022YFA1204100,2018YFA0305800,2019YFA0308500,2020YFA0308800,and 2022YFA1403400)+4 种基金the CAS Project for Young Scientists in Basic Research(YSBR-003 and 2022YSBR-048)the Innovation Program of Quantum Science and Technology(2021ZD0302700)the financial support from the European Research Council(ERC Consolidator Grant “Nonlinear Topo”,No.815869)ISF-Singapore-Israel Research Grant(3520/20)supported by the US DOE,Basic Energy Sciences(DE-FG02-99ER45747)。
文摘Vortices and bound states offer an effective means of comprehending the electronic properties of superconductors.Recently,surface-dependent vortex core states have been observed in the newly discovered kagome superconductors CsV_(3)Sb_(5).Although the spatial distribution of the sharp zero energy conductance peak appears similar to Majorana bound states arising from the superconducting Dirac surface states,its origin remains elusive.In this study,we present observations of tunable vortex bound states(VBSs)in two chemically-doped kagome superconductors Cs(V_(1-x)Tr_(x))_(3)Sb_(5)(Tr=Ta or Ti),using low-temperature scanning tunneling microscopy/spectroscopy.The CsV_(3)Sb_(5)-derived kagome superconductors exhibit full-gap-pairing superconductivity accompanied by the absence of long-range charge orders,in contrast to pristine CsV_(3)Sb_(5).Zero-energy conductance maps demonstrate a field-driven continuous reorientation transition of the vortex lattice,suggesting multiband superconductivity.The Ta-doped CsV_(3)Sb_(5)displays the conventional cross-shaped spatial evolution of Caroli-de Gennes-Matricon bound states,while the Tidoped CsV_(3)Sb_(5)exhibits a sharp,non-split zero-bias conductance peak(ZBCP)that persists over a long distance across the vortex.The spatial evolution of the non-split ZBCP is robust against surface effects and external magnetic field but is related to the doping concentrations.Our study reveals the tunable VBSs in multiband chemically-doped CsV_(3)Sb_(5)system and offers fresh insights into previously reported Y-shaped ZBCP in a non-quantum-limit condition at the surface of kagome superconductor.
基金This study was supported by the National Natural Science Foundation of China(No.12102027).
文摘Experimental folding fin models with an adjustable free-play are tested in a wind tunnel.The fin structure is modeled using the free-interface component mode synthesis method,and its free-play is modeled as four independent nonlinear springs with asymmetric stiffness.A nonplanar unsteady vortex-lattice method considering compressibility is employed to address nonlinear deformation and high subsonic flow.Surface spline interpolation is improved through projection and partition.The aeroelastic characteristics of folding fins with different free-play magnitudes,initial conditions and elastic-axis positions are analyzed using an established time-marching method because of its relatively small computation scale and high precision.The results show good consistency among the presented method,the wind tunnel test and the harmonic balance method.There is a negative correlation between the critical speed of divergent motion and the ratio of the initial condition to the free-play magnitude.If either the free-play magnitude or the initial condition is extreme(tiny or vast),the system nonlinearity degenerates to linearity.Generally,the flutter prevention design of a linear model can be applied to a nonlinear model,such as moving the elastic-axis position aftward.The presented fin configuration exhibits an unstable limit cycle oscillation because the orders of coupled flutter modes do not change with variations in equivalent linear stiffness.
文摘In this work, we study approximations of supercritical or suction vortices in tornadic flows and their contribution to tornadogenesis and tornado maintenance using self-avoiding walks on a cubic lattice. We extend the previous work on turbulence by A. Chorin and collaborators to approximate the statistical equilibrium quantities of vortex filaments on a cubic lattice when both an energy and a statistical temperature are involved. Our results confirm that supercritical (smooth, “straight”) vortices have the highest average energy and correspond to negative temperatures in this model. The lowest-energy configurations are folded up and “balled up” to a great extent. The results support A. Chorin’s findings that, in the context of supercritical vortices in a tornadic flow, when such high-energy vortices stretch, they need to fold and transfer energy to the surrounding flow, contributing to tornado maintenance or leading to its genesis. The computations are performed using a Markov Chain Monte Carlo approach with a simple sampling algorithm using local transformations that allow the results to be reliable over a wide range of statistical temperatures, unlike the originally used pivot algorithm that only performs well near infinite temperatures. Efficient ways to compute entropy are discussed and show that a system with supercritical vortices will increase entropy by having these vortices fold and transfer their energy to the surrounding flow.
文摘We present an overview of our recent theoretical studies on the quantum phenomena of the spin-1 Bose Einstein condensates, including the phase diagram, soliton solutions and the formation of the topological spin textures. A brief exploration of the effects of spin-orbit coupling on the ground-state properties is given. We put forward proposals by using the transmission spectra of an optical cavity to probe the quantum ground states： the ferromagnetic and polar phases. Quasi-one-dimension solitons and ring dark solitons are studied. It is predicted that characteristics of the magnetic solitons in optical lattice can be tuned by controlling the long-range light-induced and static magnetic dipole- dipole interactions; solutions of single-component magnetic and single-, two-, three-components polar solitons are found; ring dark solitons in spin-1 condensates are predicted to live longer lifetimes than that in their scalar counterparts. In the formation of spin textures, we have considered the theoretical model of a rapidly quenched and fast rotating trapped spin-1 Bose Einstein condensate, whose dynamics can be studied by solving the stochastic projected Gross-Pitaevskii equations. Spontaneous generation of nontrivial topological defects, such as the hexagonal lattice skyrmions and square lattice of half-quantized vortices was predicted. In particular, crystallization of merons （half skyrmions） can be generated in the presence of spin-orbit coupling.
基金supported by National Key R&D Program of China(Grant No.2022YFA1403201)National Natural Science Foundation of China(Grant No.12274205 and No.11874205).
基金the National Council for Science and Technology of Mexico(Consejo Nacional de Ciencia y Tecnología-CONACyT).
文摘One of the main issues concerning the standard Vortex Lattice Method is its application to partially or fully detached flow conditions,where non-linear aerodynamic characteristics appear as the angle of attack increases and/or the aspect ratio decreases.In order to solve such limitations,a pure numerical approach based entirely on the Vortex Lattice Method concepts has been developed.The so-called steady“Full Multi-wake Vortex Lattice Method”comes from the main hypothesis that each discretized element on the body’s surface detaches their own wakes downstream.The obtained results match for lift,drag and moment coefficients for the entire aspect ratio range configurations(under straight wakes and inviscid assumptions).Future unsteady versions of such a multi-wake approach could improve the current results obtained through Vortex Element Methods(as vortons or isolated vortex filaments).
基金supported by the National Key R&D Program of China(Grant Nos.2018YFA0307500 and 2023YFA1407100)the Key Scientific and Technological Innovation Team of Shaanxi Province(Grant No.2021TD-56)+7 种基金the National Natural Science Foundation of China(Grant Nos.12074303,62022066,12074306,and 11804267)the IBS Young Scientist Fellowship(Grant No.IBS-R024-Y3)the Basis Foundation(Grant No.21-1-3-30-1)the support of the European Union’s Horizon 2020 program,through an FET Open research and innovation action(Grant No.964770)(Topo Light)he ANR projects Labex Ga NEXT(Grant No.ANR-11-LABX0014)“NEWAVE”(Grant No.ANR-21-CE24-0019)the ANR program“Investissements d’Avenir”through the IDEX-ISITE initiative 16-IDEX-0001(Grant No.CAP 20-25)support by the Russian Science Foundation(Grant No.22-12-00144)
文摘Engineering of the orbital angular momentum(OAM)of light due to interaction with photonic lattices reveals rich physics and motivates potential applications.We report the experimental creation of regularly distributed quantized vortex arrays in momentum space by probing the honeycomb and hexagonal photonic lattices with a single focused Gaussian beam.For the honeycomb lattice,the vortices are associated with Dirac points.However,we show that the resulting spatial patterns of vortices are strongly defined by the symmetry of the wave packet evolving in the photonic lattices and not by their topological properties.Our findings reveal the underlying physics by connecting the symmetry and OAM conversion and provide a simple and efficient method to create regularly distributed multiple vortices from unstructured light.
基金The National Research Foundation of Korea(NRF-2017-R1A5A1015311 and 2021R1C1C1010198),South Korea.Author information。
文摘Electric vertical take-off and landing(eVTOL)aircraft with multiple lifting rotors or prop-rotors have received significant attention in recent years due to their great potential for next-generation urban air mobility(UAM).Numerical models have been developed and validated as predictive tools to analyze rotor aerodynamics and wake dynamics.Among various numerical approaches,the vortex method is one of the most suitable because it can provide accurate solutions with an affordable computational cost and can represent vorticity fields downstream without numerical dissipation error.This paper presents a brief review of the progress of vortex methods,along with their principles,advantages,and shortcomings.Applications of the vortex methods for modeling the rotor aerodynamics and wake dynamics are also described.However,the vortex methods suffer from the problem that it cannot deal with the nonlinear aerodynamic characteristics associated with the viscous effects and the flow behaviors in the post-stall regime.To overcome the intrinsic drawbacks of the vortex methods,recent progress in a numerical method proposed by the authors is introduced,and model validation against experimental data is discussed in detail.The validation works show that nonlinear vortex lattice method(NVLM)coupled with vortex particle method(VPM)can predict the unsteady aerodynamic forces and complex evolution of the rotor wake.
文摘This paper presents the application of an artificial neural network to develop an approach to determine and study the energy-optimal wing kinematics of a hovering bionic hawkmoth model.A three-layered artificial neural network is used for the rapid prediction of the unsteady aerodynamic force acting on the wings and the required power.When this artificial network is integrated into genetic and simplex algorithms,the running time of the optimization process is reduced considerably.The validity of this new approach is confirmed in a comparison with a conventional method using an aerodynamic model based on an extended unsteady vortex-lattice method for a sinu soidal wing kinematics problem.When studying the obtained results,it is found that actual hawkmoths do not hover under an energy-optimal condition.Instead,by tilting the stroke plane and lowering the wing positions,they can compromise and expend some energy to enhance their maneuverability and the stability of their flight.
基金Acknowledgements M.R. Eskildsen was supported by the U. S. National Science Foundation through grant DMR-0804887. Collaboration and stimulating discussions are acknowledged with： A. B. Abrahamsen, T. M. Artemova, E. D. Bauer, A. D. Bianchi, T. D. Blasius, S. L. Bud＇ko, P. C. Canfield, P. Das, L. DeBeer-Schmitt, J. M. Densmore, C. D. Dewhurst, Z. Fisk, E. M. Forgan, J. L. Gavilano, S. Gerber, A. I. Coldman, M. Ichioka, R. Ikeda, N. Jenkins, M. Kenzelmann, V. G. Kogan, J. Kohlbrecher, A. Kreyssig, M. Laver, K. Machida, J. Mesot, R. Movshovich, N. Ni, T. O＇Brien, C. Petrovic, R. Prozorov, K. Rovira, J. L. Sarrao, I. S. Veshchunov, L. Ya. Vinnikov. J. S. White and M. Zolliker.
文摘Here we review recent small-angle scattering studies of the vortex lattice in a range of type-II superconductors carried out by our group. Emphasis is placed on providing examples of the kind of information which can be obtained by such measurements, focusing in particular on studies of the vortex lattice structure and form factor in LuNi2B2C, TmNi2B2C, CeCoIn5 and Ba（Fe0.93Co0.07）2As2.
文摘The standard Ginzburg-Landau (GL) equations are only valid in the vicinity of the critical temperature. Based on the Eilenberger equations for a single band and s-wave superconductor, we derive a modified version of the standard GL equations to improve the applicability of the standard formalism at temperature away from the critical temperature. It is shown that in comparison with previous studies, our method is more convenient to calculate and our modified equations are also compatible with a dirty superconductor. To illustrate the usefulness of our formalism, we solve the modified equations numerically and give the magnetic field distribution in the mixed state at any temperature. The results show that the vortex lattice could be still observed even away from the critical temperature (e.g., T/Tc = 0.3).
